# The edge-disjoint paths problem in Eulerian graphs and 4-edge-connected graphs

## Abstract

In the edge-disjoint paths problem, we are given a graph and a set of k pairs of vertices, and we have to decide whether or not the graph has k edge-disjoint paths connecting given pairs of terminals. Robertson and Seymour’s graph minor project gives rise to a polynomial time algorithm for this problem for any fixed k, but their proof of the correctness needs the whole Graph Minor project. We give a faster algorithm and a much simpler proof of the correctness for the edge-disjoint paths problem. Our results can be summarized as follows:

1. 1.

If an input graph is either 4-edge-connected or Eulerian, then our algorithm only needs to look for the following three simple reductions: (i) Excluding vertices of high degree. (ii) Excluding ≤3-edge-cuts. (iii) Excluding large clique minors.

2. 2.

When an input graph is either 4-edge-connected or Eulerian, the number of terminals k is allowed to be a non-trivially super constant number, up to k=O((log log logn)½−ε) for any ε > 0. In addition, if an input graph is either 4-edge-connected planar or Eulerian planar, k is allowed to be O((logn ½−ε) for any ε > 0.

3. 3.

We also give our own algorithm for the edge-disjoint paths problem in general graphs. We basically follow the Robertson-Seymour’s algorithm, but we cut half of the proof of the correctness for their algorithm. In addition, our algorithm is faster than Robertson and Seymour’s.

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## Author information

Authors

### Corresponding author

Correspondence to Yusuke Kobayashi.

An extended abstract of this paper appears in SODA 2010 [14].

Research partly supported by Japan Society for the Promotion of Science, Grant-in-Aid for Scientific Research, by C & C Foundation, by Kayamori Foundation and by Inoue Research Award for Young Scientists.

JST, ERATO, Kawarabayashi Large Graph Project, Japan.

Supported by the Grant-in-Aid for Scientific Research, MEXT, Japan.

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Kawarabayashi, Ki., Kobayashi, Y. The edge-disjoint paths problem in Eulerian graphs and 4-edge-connected graphs. Combinatorica 35, 477–495 (2015). https://doi.org/10.1007/s00493-014-2828-6